On quasi-duo rings

Abstract

1. Introduction. Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass' conjecture is true for left duo rings. Call a ring R weakly left duo if for every r e R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rr" {r) is two-sided. Recently, Xue [21] proved that Bass' conjecture is still true for weakly left duo rings.In Section 3 of this paper, we first generalize the above results to left quasi-duo rings. We call a ring R a left (right) quasi-duo ring if every maximal left (right) ideal of R is two-sided. It is shown that the class of all weakly left duo rings is properly contained in the class of all left quasi-duo rings but that several basic properties of weakly left duo rings as proved in [22] are valid for left quasi-duo rings. This is the content of Section 2. Furthermore, we establish in Section 4 the equivalence of the following conditions on a left quasi-duo ring R: (1) R is a left P-exchange ring; (2) Throughout this paper all rings are associative with identity and modules are unitary left modules unless otherwise specified. J(R) always denotes the Jacobson radical of a ring R. Homomorphisms of modules will be written on the side of their arguments opposite to scalars.

g.x/-full clean rings

“…Yu [5] called a ring R to be a left quasi-duo ring if every maximal left ideal of R is a two-sided ideal. Commutative rings, local rings, rings in which every nonunit has a power that is central are all belong to this class of rings [5]. Theorem 1.…”

Finite homomorphic images of Bezout duo-domains

“…Here we need to mention that the quasi-duo rings have been studied in [6,15], where the reader can find the proofs of their basic properties and their connections to the classes of regular and exchange rings. Furthermore, for the Bezout rings (as well as the arithmetical rings) the quasi-duo conditions have tight connection to the right distributivity of lattice of its right ideals.…”

Pierce Stalks of Exchange Rings

“…Let A ∈ M n (R). Since R is a right (left) quasi-duo exchange ring, by [12,Lemma 2.3], R/J(R) is a normal exchange ring. In view of Theorem 4.6, there exists U ∈ GL n R/J(R) such that A ± U ∈ GL n R/J(R) .…”